Praise be to Allah.
If the University or the body that gives this bursary stipulates that it should be spent on matters having to do with study, then he has to adhere to that stipulation and not spend it on anything else, even if it is for Hajj.
If the body that granted this bursary did not make any such stipulation; rather it gave it to him because he is a student, then it has become his property on the basis that he is entitled to it, and he may spend it on whatever he wants of permissible things or save it for Hajj and take a student loan to spend on his maintenance.
But that is on condition that this loan is not based on riba and does not involve paying back more than he took, even if the additional amount is very small, because that is riba which is haraam.
But we do not advise the questioner to get involved in debt, even with the aim of doing Hajj; rather it is not permissible for him to get into this loan in the first place if he does not have the means to pay it back and does not have any hope of doing so.
If he does hope that he will be able to pay back, there is nothing wrong with him taking it on, so long as it is free of riba. But it is better for a person not to take debt upon himself whenever it is possible to avoid that.
And Allah knows best.